The thermodynamic limit of the Lieb-Liniger model was first investigated in [1]}. For a generic state, we can approximate the distribution of momenta by a spectral, or phase space, density \(\rho _{\rm p}(p)\) , which is such that \(\rho _{\rm p}(p_i) \approx (L(p_{i+1}-p_i))^{-1}\) . Defining a density of holes by \(\rho _{\rm h}(p_i)= (I_{i+1}-I_i)\rho _{\rm p}(p_i)\) , the Bethe equations (REF ) become [2]}
\(2\pi \rho _{\rm s}(p)= P^{\prime }(p) - \int {\rm d}p^{\prime }\,\varphi (p,p^{\prime })\rho _{\rm p}(p^{\prime }) \; ,\) 
